NBER WORKING PAPER SERIES VALUATION IN OVER-THE-COUNTER MARKETS. Darrell Duffie Nicolae Gârleanu Lasse Heje Pedersen

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1 NBER WORKING PAPER SERIES VALUATION IN OVER-THE-COUNTER MARKETS Darrell Duffie Nicolae Gârleanu Lasse Heje Pedersen Working Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA February 2006 This paper includes work previously distributed under the title Valuation in Dynamic Bargaining Markets. We are grateful for an insightful comment of Romans Pancs, for conversations with Yakov Amihud, Helmut Bester, Joseph Langsam, Richard Lyons, Tano Santos, and Jeff Zwiebel, and to participants at the NBER Asset Pricing Meeting, the Cowles Foundation Incomplete Markets and Strategic Games Conference, Hitotsubashi University, The London School of Economics, The University of Pennsylania, the Western Finance Association conference, the CEPR meeting at Gerzensee, University College London, The University of California, Berkeley, Universite Libre de Bruxelles, Tel Aviv University, Yale University, and Universitat Autonoma de Barcelona. We also thank Gustavo Manso for research assistance, as well as the editor and referees for helpful suggestions. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research by Darrell Duffie, Nicolae Gârleanu, and Lasse Heje Pedersen. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including notice, is given to the source.

2 Valuation in Over-the-Counter Markets Darrell Duffie, Nicolae Gârleanu, and Lasse Heje Pedersen NBER Working Paper No February 2006 JEL No. G0, G1, G12 ABSTRACT We provide the impact on asset prices of search-and-bargaining frictions in over-the-counter markets. Under certain conditions, illiquidity discounts are higher when counterparties are harder to find, when sellers have less bargaining power, when the fraction of qualified owners is smaller, or when risk aversion, volatility, or hedging demand are larger. Supply shocks cause prices to jump, and then recover over time, with a time signature that is exaggerated by search frictions. We discuss a variety of empirical implications. Darrell Duffie Graduate School of Business Stanford University Stanford, CA and NBER duffie@stanford.edu Nicolae Gârleanu Wharton School University of Pennsylvania 3620 Locust Walk Philadelphia, PA garleanu@wharton.upenn.edu Lasse Heje Pedersen NYU Stern Finance 44 West Fourth Street Suite New York, NY and NBER lpederse@stern.nyu.edu

3 Many assets, such as mortgage-backed securities, corporate bonds, government bonds, emerging-market debt, bank loans, swaps and many other derivatives, private equity, and real estate, are traded in over-the-counter (OTC) markets. Traders in these markets search for counterparties, incurring opportunity or other costs. When counterparties meet, their bilateral relationship is strategic; prices are set through a bargaining process that reflects each investor s alternatives to immediate trade. We provide a theory of dynamic asset pricing that directly treats search and bargaining in OTC markets. We show how the explicitly calculated equilibrium allocations and prices depend on investors search abilities, bargaining powers, and risk aversion, and how the time signature of price reactions to supply or demand shocks depends on the speed with which counterparties interact. We discuss a variety of financial applications and testable implications. Investors in our model contact one another randomly at some mean rate that reflects search ability. When two agents meet, they bargain over the terms of trade based on endogenously determined outside options. Investors are infinitely lived and gains from trade arise from time-varying costs or benefits of holding assets. We show how the equilibrium bargaining powers of the counterparties are determined by search opportunities, using the approach of Rubinstein and Wolinsky (1985). We first study how search frictions affect asset prices in a steady-state equilibrium in which agents face idiosyncratic risk and with no aggregate risk. We compute steady-state prices both with risk-neutral and risk-averse agents, and show how risk aversion can be approximated in a risk-neutral setting using holding costs that capture the utility losses of suboptimal diversification or hedging. Naturally, search-based market incompleteness is priced by risk-averse agents with time-varying hedging demands. Indeed, under stated conditions, illiquidity discounts are higher if investors can find each other less easily, if buyers have more bargaining power, if the fraction of qualified owners is smaller, if volatility is higher, or if risk aversion is higher. We also indicate situations in which search frictions can lead naturally to an increase in the price of the asset, conveying to it a search-induced scarcity value. We introduce aggregate liquidity shocks, that is, shocks that affect the holding costs of many agents simultaneously. We show that, under certain conditions, when an aggregate liquidity shock occurs, the price drops and recovers only slowly. The speed of the price recovery depends on the search 2

4 intensity that determines the speed of reallocation of securities to the more liquid agents and on the time that it takes for illiquid agents to become liquid, for example to recapitalize. Also, the risk of future aggregate liquidity shocks significantly lowers the post-recovery price level. Search frictions thus affect both the general level of prices as well as the resiliency of the market to aggregate shocks. More illiquid markets (those with lower search intensities) have generally lower price levels, larger price reactions to supply shocks, and slower price recovery. When an aggregate liquidity shock occurs, the expected utilities of asset owners, even those owners who are not directly affected by the liquidity shock themselves, decrease because selling opportunities worsen: Sellers search times increase and their bargaining positions deteriorate. Conversely, the expected utilities of agents waiting on the sideline, those with no asset position, increase at times of aggregate liquidity shock, because they may have the opportunity to purchase securities at distressed prices. We discuss how our results contribute to an explanation of the time signatures of price responses to several types of aggregate liquidity shocks, to corporate bonds that are downgraded or in default (Hradsky and Long (1989)), to sovereign bonds during debt crises, to individual stocks at index inclusion or exclusion events (as in Greenwood (2005)), to stocks affected by sudden large outside orders (Andrade, Chang, and Seasholes (2005) and Coval and Stafford (2005)), or to catastrophe reinsurance risk premia after large unexpected losses in capital caused by events such as major hurricanes (Froot and O Connell (1999)), among other relevant empirical phenomena. The point of departure of this paper is a variant of the basic risk-neutral search-based pricing model of Duffie, Gârleanu, and Pedersen (2005). While Duffie, Gârleanu, and Pedersen (2005) focus on the steady-state pricing of a simple consol bond and treat the behavior of marketmakers and the implications of search frictions for bid-ask spreads, this paper instead treats the implications of search frictions for risky asset pricing. We provide (i) the impact on asset prices of risk aversion in a setting with search, above and beyond the usual implications of risk sharing in incomplete markets, (ii) the implications of search frictions for the time dynamics of price responses to supply or demand shocks, and (iii) the determination of endogenous bargaining power 1 based on the alternative search opportunities of the buyers and 1 Duffie, Gârleanu, and Pedersen (2005) take the bargaining powers of buyers and sellers as exogenous to the model. 3

5 sellers. Search models have been studied extensively in the context of labor economics, starting with the coconuts model of Diamond (1982), and in the context of monetary economics, for example, Trejos and Wright (1995). As for search-based models of asset pricing, Weill (2002) and Vayanos and Wang (2002) have extended the risk-neutral version of our model in order to treat multiple assets in the same economy, obtaining cross-sectional restrictions on asset returns. Duffie, Gârleanu, and Pedersen (2005) treat marketmakers, showing that search frictions have different implications for bid-ask spreads than do information frictions. Miao (2004) provides a variant of this model. Weill (2003) studies the implications of search frictions in an extension of our model in which marketmakers inventories lean against the outside order flow. Newman and Rierson (2003) present a model in which supply shocks temporarily depress prices across correlated assets, as providers of liquidity search for long-term investors, supported by empirical evidence of issuance impacts across the European telecommunications bond market. Duffie, Gârleanu, and Pedersen (2002) use a search-based model of the impact on asset prices and securities lending fees of the common institution by which would-be shortsellers must locate lenders of securities before being able to sell short. Difficulties in locating lenders of shares can allow for dramatic price imperfections, as, for example, in the case of the spinoff of Palm, Incorporated, documented by Mitchell, Pulvino, and Stafford (2002) and Lamont and Thaler (2003). Our results also complement the literature treating the effect on asset prices of an exogenously specified trading cost (Amihud and Mendelson (1986), Constantinides (1986), Vayanos (1998), Huang (2003), and Acharya and Pedersen (2005)) by endogenizing the trading cost in the context of OTC markets. Krainer and LeRoy (2002) study housing prices in a different search framework. Longstaff (2004) addresses market frictions with the device of deterministic blackout periods on individual trade. The remainder of the paper is organized as follows. Section 1 lays out a baseline model with risk-neutral agents. Section 2 then treats an OTC market for a risky asset whose risk-averse owners search for potential buyers when the asset ceases to be a relatively good endowment hedge. We characterize how search frictions magnify risk premia beyond those of a liquid but incompletemarkets setting. Section 3 provides the implications of search frictions for price reactions to supply or demand shocks, showing especially how the time pattern of price recovery is influenced by search frictions. Finally, Section 4 4

6 describes the empirical implications of search frictions for asset pricing in a range of actual OTC markets. Some proofs and supplementary results are relegated to appendices. 1 Basic Search Model of Asset Prices This section introduces a baseline risk-neutral model of an over-the-counter market, that is, a market in which agents can trade only when they meet each other, and in which transaction prices are bargained. This baseline model, simplified from Duffie, Gârleanu, and Pedersen (2005) by stripping out marketmakers, is then generalized in the remainder of the paper to treat risk aversion, and the effects of aggregate liquidity shocks. Agents are risk-neutral and infinitely lived, with a constant time-preference rate β > 0 for consumption of a single non-storable numeraire good. 2 An agent can invest in a bank account which can also be interpreted as a liquid security with a risk-free interest rate of r. As a form of credit constraint that rules out Ponzi schemes, the agent must enforce some lower bound on the liquid wealth process W. We take r = β in this baseline model, since agents are risk neutral. Agents may trade a long-lived asset in an over-the-counter market. The asset can be traded only bilaterally, when in contact with a counterparty. We begin for simplicity by taking the OTC asset to be a consol, which pays one unit of consumption per unit of time. Later, when introducing the effect of risk aversion, we generalize to random dividend processes. An agent is characterized by an intrinsic preference for asset ownership that is high or low. A low-type agent, when owning the asset, has a holding cost of δ per time unit. A high-type agent has no such holding cost. We could imagine this holding cost to be a shadow price for ownership due, for example, to (i) low liquidity, that is, a need for cash, (ii) high financing or financial-distress costs, (iii) adverse correlation of asset returns with endowments (formalized in Section 2), (iv) a relative tax disadvantage, as studied by Dai and Rydqvist (2003) in an empirical analysis of searchand-bargaining effects in the context of tax trading, 3 or (v) a relatively low 2 Specifically, an agent s preferences among adapted finite-variation cumulative consumption processes are represented by the utility E ( ) e βt dc 0 t for a cumulative consumption process C, whenever the integral is well defined. 3 Dai and Rydqvist (2003) study tax trading between a small group of foreign investors 5

7 personal use for the asset, as may happen, for example, for certain durable consumption goods such as homes. The agent s intrinsic type is a Markov chain, switching from low to high with intensity λ u, and back with intensity λ d. The intrinsic-type processes of any two agents are independent. 4 A fraction s of agents are initially endowed with one unit of the asset. Investors can hold at most one unit of the asset and cannot shortsell. Because agents have linear utility, it is without much loss of generality that we restrict attention to equilibria in which, at any given time and state of the world, an agent holds either 0 or 1 unit of the asset. Hence, the full set of agent types is T = {ho, hn, lo, ln}, with the letters h and l designating the agent s current intrinsic liquidity state as high or low, respectively, and with o or n indicating whether the agent currently owns the asset or not, respectively. We suppose that there is a continuum (a non-atomic finite measure space) of agents, and let µ σ (t) denote the fraction at time t of agents of type σ T, so that 1 = µ ho (t) + µ hn (t) + µ lo (t) + µ ln (t). (1) Equating the per-capita supply s with the fraction of owners gives s = µ ho (t) + µ lo (t). (2) An agent finds a counterparty with an intensity λ, reflecting the efficiency of the search technology. We assume the counterparty found is randomly selected from the pool of other agents, so that the probability that the counterparty is of type σ is µ σ (t). Thus, the total intensity of finding a type-σ investor is λµ σ. Hence, assuming that the law of large numbers applies, hn investors contact lo investors at a total (almost sure) rate of λµ lo µ hn and, since lo investors contact hn investors at the same total rate, the total rate of such counterparty matchings is 2λµ lo µ hn. Duffie and Sun (2004) provide and a larger group of domestic investors. They find that investors from the long side of the market get part of the gains from trade, under certain conditions, which they interpret as evidence of a search-and-bargaining equilibrium. 4 All random variables are defined on a probability space (Ω, F, Pr) with corresponding filtration {F t : t 0} of sub-σ-algebras of F satisfying the usual conditions, as defined by Protter (1990). The filtration represents the resolution over time of information commonly available to investors. 6

8 a discrete-time search-and-matching model in which the exact law of large numbers for a continuum of agents indeed applies in this sense. 5 To solve the model, we proceed in two steps. First, we use the insight that the only form of encounter that provides gains from trade is one in which low-type owners sell to high-type non-owners. From bargaining theory, we know (see Appendix A) that at these encounters, trade occurs immediately. We can therefore determine the asset allocations without reference to prices. Given the time-dynamics of the masses, {µ(t) : t 0}, we then consider an investor s lifetime utility, depending on the investor s type, the bargaining problem, and the resulting price. In equilibrium, the rates of change of the fractions of the respective investor types are µ lo (t) = 2λµ hn (t)µ lo (t) λ u µ lo (t) + λ d µ ho (t) µ hn (t) = 2λµ hn (t)µ lo (t) λ d µ hn (t) + λ u µ ln (t) µ ho (t) = 2λµ hn (t)µ lo (t) λ d µ ho (t) + λ u µ lo (t) (3) µ ln (t) = 2λµ hn (t)µ lo (t) λ u µ ln (t) + λ d µ hn (t). The intuition for, say, the first equation in (3) is straightforward: Whenever an lo agent meets an hn investor, he sells his asset and is no longer an lo agent. This (together with the law of large numbers) explains the first term on the right hand side of (3). The second term is due to intrinsic type changes in which lo investors become ho investors, and the last term is due to intrinsic type changes from ho to lo. Duffie, Gârleanu, and Pedersen (2005) show that there is a unique stable steady-state solution for {µ(t) : t 0}, that is, a constant solution defined by µ(t) = 0. The steady state is computed by using (1) (2) and the fact that µ lo + µ ln = λ d /(λ u + λ d ), in order to write the first equation in (3) as a quadratic equation in µ lo, given as Appendix equation (C.1). Having determined the steady-state fractions of investor types, we compute the investors equilibrium intensities of finding counterparties of each type and, hence, their utilities for remaining lifetime consumption, as well as the bargained price P. The utility of a particular agent depends on his current type, σ(t) T, and the wealth W(t) in his bank account. Specifically, 5 Giroux (2005) proves that the cross-sectional distribution of agent types in a natural discrete-time analogue of this model indeed converges to the continuous-time model studied here. 7

9 lifetime utility is W(t) + V σ(t), where, for each investor type σ in T, V σ is a constant to be determined. In steady state, the rate of growth of any agent s expected indirect utility must be the discount rate r, which yields the steady-state equations 0 = rv lo λ u (V ho V lo ) 2λµ hn (P V lo + V ln ) (1 δ) 0 = rv ln λ u (V hn V ln ) 0 = rv ho + λ d (V ho V lo ) 1 (4) 0 = rv hn + λ d (V hn V ln ) 2λµ lo (V ho V hn P). The price is determined through bilateral bargaining. A high-type nonowner pays at most his reservation value V h = V ho V hn for obtaining the asset, while a low-type owner requires a price of at least V l = V lo V ln. Nash bargaining, or the Rubinstein-type game considered in Appendix A, implies that the bargaining process results in the price P = V l (1 q) + V h q, (5) where q [0, 1] is the bargaining power of the seller. While Nash equilibrium is consistent with exogenously assumed bargaining powers, Appendix A applies the device of Rubinstein and Wolinsky (1985) to calculate the unique bargaining powers that represent the limiting prices of a sequence of economies in which, once a pair of counteparties meets to negotiate, one of the pair is selected at random to make an offer to the other, at each of a sequence of offer times separated by intervals that shrink to zero. Specifically, suppose that when two agents find each other, one of them is chosen randomly, the seller with probability ˆq and the buyer with probability 1 ˆq, to suggest a trading price. The other either rejects or accepts the offer, immediately. If the offer is rejected, the owner receives the dividend from the asset during the current period. At the next period, t later, one of the two agents is chosen at random, independently, to make a new offer. The bargaining may, however, break down before a counteroffer is made. A breakdown may occur because at least one of the agents changes valuation type, or if one of the agents meets yet another agent, and leaves his current trading partner, provided agents can indeed continue to search while engaged in negotiation. In that case, as shown in Appendix A, the limiting price as t goes to zero is represented by (5), with the bargaining power of the seller q equal to ˆq. This simple solution, in which the only bargaining advantage that matters in the limit is the likelihood of being selected as the agent that 8

10 makes the next offer, arises because a counterparty s ability to meet an alternative trading partner while negotiating makes that counterparty more impatient, but also increases the trading partner s risk of breakdown, to the point that these two effects are precisely offseting. If, however, agents cannot search for alternative trading partners during negotiations, then the limiting price is that associated with the bargaining power q = ˆq(r + λ u + λ d + 2λµ lo ) ˆq(r + λ u + λ d + 2λµ lo ) + (1 ˆq)(r + λ u + λ d + 2λµ hn ). (6) For the comparative statics that follow, we will use the limiting bargaining power associated with search during negotiation, in order to simplify the analysis by avoiding the dependence in (6) of the seller s bargaining power q on various parameters that may be shifted as part of the experiment being considered. The linear system (4)-(5) of equations has a unique solution, with P = 1 r δ r r(1 q) + λ d + 2λµ lo (1 q) r + λ d + 2λµ lo (1 q) + λ u + 2λµ hn q. (7) This price (7) is the present value, 1/r, of dividends, reduced by an illiquidity discount. The price is lower and the discount is larger, ceteris paribus, if the distressed owner has less hope of switching type (lower λ u ), if it is more difficult for the owner to find other buyers (lower µ hn ), if the buyer may more suddenly need liquidity himself (higher λ d ), if it is easier for the buyer to find other sellers (higher µ lo ), or if the seller has less bargaining power (lower q). These intuitive results are based on partial derivatives of the right-hand side of (7) in other words, they hold when a parameter changes without influencing any of the others. We note, however, that the steady-state type fractions µ themselves depend on λ d, λ u, and λ. The following proposition offers a characterization of the equilibrium steady-state effect of changing each parameter. Proposition 1 The steady-state equilibrium price P is decreasing in δ, s, and λ d, and is increasing in λ u and q. Further, if s < λ u /(λ u + λ d ), then P 1/r as λ, and P is increasing in λ for all λ λ, for a constant λ depending on the other parameters of the model. 9

11 The condition that s < λ u /(λ u + λ d ) means that, in steady state, there is less than one unit of asset per agent of high intrinsic type. While this corresponds to the intuitively anticipated increase in market value with increasing bilateral contact rate, the alternative is also possible. With s > λ u /(λ u +λ d ), the marginal investor in perfect markets has the relatively lower reservation value, and search frictions lead to a scarcity value. For example, a hightype investor in an illiquid OTC market could pay more than the Walrasian price for the asset because it is hard to find, and given no opportunity to exploit the effect of immediate competition among many sellers. This scarcity value could, for example, contribute to the widely studied on-the-run premium for Treasuries, as discussed in Section 4. It can be checked that the above results extend to risky dividends in at least the following senses: (i) If the cumulative dividend is risky with constant drift ν, then the equilibrium is that for a consol bond with dividend rate of ν; (ii) if the dividend rate and illiquidity cost are proportional to a process X with E t [X(t + u)] = X(t)e νu, for some constant growth rate ν, then the price and value functions are also proportional to X, with factors of proportionality given as above, with r replaced by r ν; (iii) if the dividendrate process X satisfies E t [X(t + u)] = X(t) + mu for a constant drift m (and if illiquidity costs are constant), then the continuation values are of the form X(t)/r + v σ for owners and v σ for non-owners, where the constants v σ are computed in a similar manner. Next, we model risky dividends, using case (i) above, in the context of risk-averse agents. In a previous version of the paper, we described how one can use case (iii) in a model in which investors have risk limits based on value at risk. 2 Risk-Aversion This section provides a version of the asset-pricing model with risk aversion, in which the motive for trade between two agents is the different extent to which they derive hedging benefits from owning the asset. We provide a sense in which this economy can be interpreted in terms of the baseline economy of Section 1. Agents have constant-absolute-risk-averse (CARA) additive utility, with a coefficient γ of absolute risk aversion and with time preference at rate β. 10

12 An asset has a cumulative dividend process D satisfying dd(t) = m D dt + σ D db(t), (8) where m D and σ D are constants, and B is a standard Brownian motion with respect to the given probability space and filtration (F t ). Agent i has a cumulative endowment process η i, with dη i (t) = m η dt + σ η db i (t), (9) where the standard Brownian motion B i is defined by db i (t) = ρ i (t) db(t) + 1 ρ i (t) 2 dz i (t), (10) for a standard Brownian motion Z i independent of B, and where ρ i (t) is the instantaneous correlation between the asset dividend and the endowment of agent i. We model ρ i as a two-state Markov chain with states ρ h and ρ l > ρ h. The intrinsic type of an agent is identified with this correlation parameter. An agent i whose intrinsic type is currently high (that is, with ρ i (t) = ρ h ) values the asset more highly than does a low-intrinsic-type agent, because the increments of the high-type endowment have lower conditional correlation with the asset s dividends. As in the baseline model of Section 1, agents intrinsic types are pairwise-independent Markov chains, switching from l to h with intensity λ u, and from h to l with intensity λ d. An agent owns either θ n or θ o units of the asset, where θ n < θ o. For simplicity, no other positions are permitted, which entails a loss in generality. Agents can trade the OTC security only when they meet, as previously. The agent type space is T = {lo, ln, ho, hn}. In this case, the symbols o and n indicate large and small owners, respectively. Given a total supply Θ of shares per investor, market clearing requires that (µ lo + µ ho )θ o + (µ ln + µ hn )θ n = Θ, (11) which, using (1), implies that the fraction of large owners is µ lo + µ ho = s Θ θ n θ o θ n. (12) We consider a particular agent whose type process is {σ(t) : t 0}, and let θ denote the associated asset-position process (that is, θ(t) = θ o whenever 11

13 σ(t) {ho, lo} and otherwise θ(t) = θ n ). We suppose that there is a perfectly liquid money-market asset with a constant risk-free rate of return r, which, for simplicity, is assumed to be determined outside of the model, and with a perfectly elastic supply, as is typical in the literature treating multi-period asset-pricing models based on CARA utility, such as Wang (1994). The agent s money-market wealth process W therefore satisfies dw(t) = (rw(t) c(t)) dt + θ(t) dd(t) + dη(t) P dθ(t), where c is the agent s consumption process, η is the agent s cumulative endowment process, and P is the asset price per share (which is constant in the equilibria that we examine). The last term thus captures payments in connection with trade. The consumption process is required to satisfy measurability, integrability, and transversality conditions stated in Appendix C. We consider a steady-state equilibrium, and let J(w, σ) denote the indirect utility of an agent of type σ {lo, ln, ho, hn} with current wealth w. Assuming sufficient differentiability, the Hamilton-Jacobi-Bellman (HJB) equation for an agent of current type lo is 0 = sup c R { e γc + J w (w, lo)(rw c + θ o m D + m η ) J ww(w, lo)(θ 2 o σ2 D + σ2 η + 2ρ lθ o σ D σ η ) βj(w, lo) + λ u [J(w, ho) J(w, lo)] + 2λµ hn [J(w + Pθ, ln) J(w, lo)]}, where θ = θ o θ n. The HJB equations for the other agent types are similar. Under technical regularity conditions found in Appendix C, we verify that J(w, σ) = e rγ(w+aσ+ā), (13) where ā = 1 ( log r r γ + m η 1 2 rγσ2 η r β ), (14) rγ and where, for each σ, the constant a σ is determined as follows. The firstorder conditions of the HJB equation of an agent of type σ imply an optimal consumption rate of c = log(r) γ + r(w + a σ + ā). (15) 12

14 Inserting this solution for c into the respective HJB equations yields a system of equations characterizing the coefficients a σ. The price P is determined using Nash bargaining with seller bargaining power q, similar in spirit to the baseline model of Section 1. Given the reservation values of buyer and seller implied by J(w, σ), the bargaining price satisfies a lo a ln Pθ a ho a hn. The following result obtains. Proposition 2 In equilibrium, an agent s consumption rate is given by (15), the value function is given by (13), and (a lo, a ln, a ho, a hn, P) R 5 solve 0 = ra lo + λ u e rγ(a ho a lo ) 1 rγ e rγ(a hn a ln ) 1 0 = ra ln + λ u rγ e rγ(a lo a ho ) 1 0 = ra ho + λ d rγ 0 = ra hn + λ d e rγ(a ln a hn ) 1 rγ with + 2λµ hn e rγ(pθ+a ln a lo ) 1 rγ (κ(θ o ) θ o δ) (κ(θ n ) θ n δ) (16) κ(θ o ) + 2λµ lo e rγ( Pθ+a ho a hn ) 1 rγ κ(θ) = θm D 1 2 rγ ( θ 2 σ 2 D + 2ρ h θσ D σ η ) κ(θ n ), (17) δ = rγ(ρ l ρ h )σ D σ η > 0, (18) as well as the Nash bargaining equation, ( ) ( ) q 1 e rγ(pθ (a lo a ln )) = (1 q) 1 e rγ( Pθ+a ho a hn ). (19) A natural benchmark is the limit price associated with vanishing search frictions, characterized as follows. Proposition 3 If s < µ hn + µ ho, then, as λ, P κ(θ o) κ(θ n ). (20) rθ 13

15 In order to compare the equilibrium for this model to that of the baseline model, we use the linearization e z 1 z, which leads to 0 ra lo λ u (a ho a lo ) 2λµ hn (Pθ a lo + a ln ) (κ(θ o ) θ o δ) 0 ra ln λ u (a hn a ln ) (κ(θ n ) θ n δ) 0 ra ho λ d (a lo a ho ) κ(θ o ) 0 ra hn λ d (a ln a hn ) 2λµ lo (a ho a hn Pθ) κ(θ n ) Pθ (1 q)(a lo a ln ) + q(a ho a hn ). These equations are of the same form as those in Section 1 for the indirect utilities and asset price in an economy with risk-neutral agents, with dividends at rate κ(θ o ) for large owners and dividends at rate κ(θ n ) for small owners, and with illiquidity costs given by δ of (18). In this sense, we can view the baseline model as a risk-neutral approximation of the effect of search illiquidity in a model with risk aversion. The approximation error goes to zero for small agent heterogeneity (that is, small ρ l ρ h ). 6 Solving specifically for the price P in the associated linear model, we have P = κ(θ o) κ(θ n ) rθ δ r(1 q) + λ d + 2λµ lo (1 q) r r + λ d + 2λµ lo (1 q) + λ u + 2λµ hn q. (21) The price is the sum of the perfect-liquidity price (that for the case of λ = + ), plus an adjustment for illiquidity that can be viewed as the present value of a perpetual stream of risk premia that are due to search frictions. The illiquidity component depends on the strength of the difference in hedging motives for trade by two types of agents, in evidence in the factor δ defined by (18). One of these types of agents can be viewed as the natural hedger; the other can be viewed as the type that provides the hedge, at an extra risk premium. The illiquidity risk premium need not be increasing in the degree of overall market risk exposure of the asset, and would be non-zero even if there were no aggregate endowment risk. 7 Graveline and McBrady 6 The error introduced by the linearization is in O ( (a ho a lo ) 2 + (a hn a ln ) 2 + (Pθ a lo + a ln ) 2), which, by continuity, is in O ( (ρ l ρ h ) 2) for a compact parameter space. Hence, if ρ l ρ h is small, then the approximation error is an order of magnitude smaller, of the order (ρ l ρ h ) 2. 7 We could arrange for the absence of aggregate endowment risk, for example by having half the population exposed positively to the asset, the other half exposed negatively, in an offsetting way, with the portions of endowment risks that are orthogonal to the asset returns being idiosyncratic and adding up (by the law of large numbers) to zero. (We 14

16 (2005) empirically link the size of repo specials in on-the-run treasuries to the motives of financial services firms to hedge their inventories of corporate and mortgage-backed securities. The repo specials, which are reflections of search frictions in the treasury repo market, are shown to be larger when the inventories are larger, and larger when interest-rate volatility is higher, consistent with (18). Numerical Example. We select parameters for a numerical illustration of the implications of the model for a market with an annual asset turnover rate of about 50%, which is roughly that of the over-the-counter market for corporate bonds. Table 1 contains the exogenous parameters for the base-case risk-neutral model, and Table 2 contains the resulting steady-state fractions of each type and the price. The search intensity of λ = 625 shown in Table 1 implies that an agent expects to be in contact with 2λ = 1250 other agents each year, that is, 1250/250 = 5 agents a day. Given the equilibrium mass of potential buyers, the average time needed to sell is 250 (2λµ hn ) 1 = 1.8 days. The switching intensities λ u and λ d mean that a high-type investor remains a high type for an average of 2 years, while an illiquid low type remains a low type for an average of 0.2 years. These intensities imply an annual turnover of 2λµ lo µ hn /s = 49% which roughly matches the median annual bond turnover of 51.7% reported by Edwards, Harris, and Piwowar (2004). The fraction of investors holding a position is s = 0.8, the discount and interest rates are 5%, sellers and buyers each have half of the bargaining power q = 0.5, and the illiquidity cost is δ = 2.5, as implied by the risk aversion parameters discussed below. We see that only a small fraction of the asset, µ lo /s = /0.8 = 0.35% of the total supply, is mis-allocated to low intrinsic types because of search frictions. The equilibrium asset price, 18.38, however, is substantially below the perfect market price of 1/r = 20, reflecting a significant impact of illiquidity on the price, despite the relatively small impact on the asset allocation. Stated differently, we can treat the asset as a bond whose yield, dividend rate of 1 divided by price, is 1/18.38 = 5.44%, or 44 basis points above the liquid-market yield r. This yield spread is of the order of magnitude of the corporate-bond liquidity spread estimated by Longstaff, Mithal, and can adjust our model so that the asset is held in zero net supply, allowing short and long positions; this was done in the risk-limits section.) 15

17 λ λ u λ d s r β q δ Table 1: Base-case parameters for baseline model. µ ho µ hn µ lo µ ln P Table 2: Steady-state masses and asset price, baseline model. Neis (2004), of between 9 and 65 basis points, depending on the specification and reference risk-free rate. The base-case risk-neutral model specified in Table 1 corresponds to a model with risk-averse agents with additional parameters given in Table 3 in the following sense. First, the illiquidity cost δ = δ = 2.5 of low-intrinsictype is that implied by (18) from the hedging costs of the risk-aversion model. Second, the total amount Θ of shares and the investor positions, θ o and θ n, imply the same fraction s = 0.8 of the population holding large positions, using (12). The investor positions that we adopt for this calibration are realistic in light of the positions adopted by high and low type investors in the associated Walrasian (perfect) market with unconstrained trade sizes, which, as shown in Appendix B, has an equilibrium large-owner position size of 17,818 shares and a small-owner position size of 2, 182 shares. Third, the certainty-equivalent dividend-rate per share, (κ(θ o ) κ(θ n ))/(θ o θ n ) = 1, is the same as that of the baseline model. Finally, the mean parameter µ D = 1 and volatility parameter σ D = 0.5 of the asset s risky dividend implies that the standard deviation of yearly returns on the bond is approximately σ D /P = 2.75%. γ ρ h ρ l µ η σ η µ D σ D Θ θ o θ n Table 3: Additional base-case parameters with risk-aversion. Figure 1 shows how prices increase with liquidity, as measured by the search intensity λ. The graph reflects the fact that, as the search intensity 16

18 70 risk neutral risk averse Price discount (%) Search intensity λ Figure 1: Proportional price reduction relative to the perfect-market price, as a function of the search intensity λ. 10 λ becomes large, the allocation and price converge to their perfect-market counterparts (Propositions 1 and 3). Figures 2 and 3 show how prices are discounted for illiquidity, relative to the perfect-markets price, by an amount that depends on risk aversion and volatility. As we vary the parameters in these figures, we compute both the equilibrium solution of the risk-aversion model and the solution of the associated baseline risk-neutral model that is obtained by the linearization (21), taking δ from (18) case by case. We see that the illiquidity discount increases with risk aversion and volatility, and that both effects are large for our benchmark parameters. The illiquidity discount ranges between 1% and 40%, depending on the risk and risk aversion. These figures also show that the equilibrium price of the OTC market model with risk aversion is generally well approximated by our closed-form expression (21). 17

19 40 35 risk neutral risk averse 30 Price discount (%) Risk aversion γ (scaled by 10 3 ) Figure 2: Proportional price reduction relative to the perfect-market price, as a function of the investor risk aversion γ. The dashed line corresponds to the model with riskaverse agents (Equations (16) (19)). The solid line corresponds to the linearized model (Equation (21)), in which the parameters δ and κ change with γ Aggregate Liquidity Shocks So far, we have studied how search frictions affect steady-state prices and returns in a setting in which agents receive idiosyncratic liquidity shocks, with no macro-uncertainty. Search frictions affect not only the average levels of asset prices, but also the asset market s resilience to aggregate shocks. We examine this by characterizing the impact of aggregate liquidity shocks that simultaneously affect many agents. We are interested in the shock s immediate effect on prices, the time-pattern of the price recovery, the ex-ante price effect due to the risk of future shocks, and the change in equilibrium search times. Our results are broadly consistent with the description by Froot and O Connell (1999) of the behavior of catastrophe reinsurance risk premia after large unexpected losses in capital caused by events such as major hurricanes. After large initial jumps in reinsurance premia caused by the sudden 18

20 25 risk neutral risk averse 20 Price discount (%) Volatility scaling factor Figure 3: Proportional price reduction relative to the perfect-market price, as a function of a volatility scaling factor that scales both σ η and σ D. The dashed line corresponds to the model with risk-averse agents (Equations (16) (19)). The solid line corresponds to the linearized model (Equation (21)), in which the parameters δ and κ change with σ η and σ D. reduction in risk-bearing capacity, gradual price declines occur over subsequent months as new capital finds its way into the sector. Chen, Noronha, and Singhal (2004), Greenwood (2005), Harris and Gurel (1986), and Kaul, Mehrotra, and Morck (2000), among others, document much shorter-lived price reactions to supply shocks in equity markets caused by changes in the definition of stock indices, at which time the position changes of committed index investors must be absorbed by other investors. Andrade, Chang, and Seasholes (2005) provide evidence of similar price reactions, again jumps followed by gradual price recoveries, following outside order imbalances on the Taiwanese stock market. While the stock markets treated in these studies are not over-the-counter, our model can be used as an abstraction of the process by which providers of liquid capital are located and drawn into the market in order to exploit temporary shortages in the levels of capital held by specialists. (Grossman and Miller (1988) and Weill (2003) focus only on 19

21 the supply of immediacy by specialists.) While highly stylized, our model of periods of abnormal expected returns and price momemtum following supply shocks also provides additional microeconomic foundations for prior asset-pricing research on limits to arbitrage, or good deals, such as Shleifer and Vishny (1997) and Cochrane and Saa-Requejo (2001). We adjust the baseline model of Section 1 (or, as explained, the linearized version of the risk-premium model of Section 2) by introducing occasional, randomly timed, aggregate liquidity shocks. At each such shock, a fraction of the agents, randomly chosen, suffer a sudden reduction in liquidity, in the sense that their intrinsic types simultaneously jump to the low state. The shocks are timed according to a Poisson arrival process, independent of all other random variables, with mean arrival rate ζ. Again appealing to the Law of Large Numbers, at each aggregate liquidity shock, the distribution of agents intrinsic types becomes µ = µ, where the post-shock distribution µ is in [0, 1] 4, satisfies (1) (2), and has an abnormally elevated quantity of illiquid agents, both owners and non-owners. Specifically, µ lo > µ lo (t) and µ ln > µ ln (t). Each high-type owner remains a high type with probability 1 π ho (t) = µ ho /µ ho (t), and becomes a low type with probability π ho. Similarly, a high-type non-owner remains high type with probability 1 π hn (t) = µ hn /µ hn (t) and becomes low type with probability π hn. Conditional on π(t), the changes in types are pairwise independent across the space of agents. This aggregate liquidity shock does not directly affect low-type agents. Of course, it affects them indirectly because of the change to the demographics of the market in which they live. By virtue of this specification, the post-shock distribution of agents does not depend on any residual aftereffects of prior shocks, a simplification without which the model would be relatively intractable. In order to solve the equilibrium with an aggregate liquidity shock, it is helpful to use the trick of measuring time in terms of the passage of time t since the last shock, rather than absolute calendar time. Knowledge of the time at which this shock occurred enables an immediate translation of the solution into calendar time. We first solve the equilibrium fractions µ(t) R 4 of agents of the four different types. At the time of an aggregate liquidity shock, this type distribution is equal to the post-shock distribution µ(0) = µ (where, to repeat, 0 means zero time units after the shock). After an aggregate liquidity shock, the cross-sectional distribution of agent types evolves according to the ODE 20

22 (3), converging (conditional on no additional shocks) to a steady state as the time since last shock increases. Given this time-varying equilibrium solution of the investor type distribution, we turn to the agents value functions. The value V σ (t) depends on the agent s type σ and the time t since the last aggregate liquidity shock. The values evolve according to V lo (t) = rv lo (t) λ u (V ho (t) V lo (t)) 2λµ hn (P(t) + V ln (t) V lo (t)) ζ(v lo (0) V lo (t)) (1 δ) V ln (t) = rv ln (t) λ u (V hn (t) V ln (t)) ζ(v ln (0) V ln (t)) V ho (t) = rv ho (t) λ d (V lo (t) V ho (t)) ζ((1 π ho (t))v ho (0) + π ho (t)v lo (0) V ho (t)) 1 (22) V hn (t) = rv hn λ d (V ln V hn ) 2λµ ho (V ho V hn P) ζ((1 π hn (t))v hn (0) + π hn (t)v ln (0) V hn ), P(t) = (V lo (t) V ln (t))(1 q) + (V ho (t) V hn (t))q, where the terms involving ζ capture the risk of an aggregate liquidity shock. This differential equation is linear in the vector V (t), depends on the deterministic evolution of µ(t), and has the somewhat unusual feature that it depends on the initial value function V (0). To solve this system, it is useful to express it in the vector form: V (t) = A 1 (µ(t))v (t) A 2 A 3 (µ(t))v (0), (23) where A 1 (µ(t)) R 4 4, A 2 R 4 1, and A 3 R 4 4 are the coefficient matrices. Treating V (0) as a fixed parameter, the unique solution to the linear ODE that satisfies the appropriate transversality condition is V (t) = At t = 0, this gives V (0) = t 0 e R s t A 1(µ(u)) du (A 2 + A 3 (µ(s))v (0)) ds. (24) e R s 0 A 1(µ(u)) du (A 2 + A 3 (µ(s))v (0)) ds, (25) and we can then derive the initial value function V (0) as the fixed point: V (0) = ( I 4 0 ) e R 1 ( ) s 0 A 1(µ(u)) du A 3 (µ(s)) ds e R s 0 A 1(µ(u)) du A 2 ds, 0 21

23 (26) where I 4 R 4 4 is the identity matrix. Equations (24) and (26) together represent the solution. One notes that we take the bargaining power q as exogenous for simplicity, rather than incorporating the effects of delay during negotiation that stem for interim changes in the value functions. Numerical Examples. We will illustrate some of the most noteworthy effects of a liquidity shock using a numerical example, and then state some general properties. We suppose that the search intensity is λ = 125, that types change idiosyncratically with intensities λ u = 2 and λ d = 0.2, that the fraction of owners is s = 0.75, that the riskless return is r = 10%, that buyers and sellers have equal bargaining powers (that is, q = 0.5), that the illiquidity loss rate is δ = 2.5, that the intensity of an aggregate liquidity shock is ζ = 0.1, and that the post-shock distribution of types is determined by µ lo = and µ ln = These parameters are consistent with a shock from steady state 8 at which high types become low types with probability 0.5. In order to motivate the results, one could imagine that an aggregate liquidity shock is associated with an event at which a large fraction of investors incur a significant loss of risk-bearing capacity, and thus have a higher shadow price for bearing risk. For example, at the default of Russia in 1998, those asset managers specializing in emerging market debt who had had positions in Russian issues would have had a substantially reduced appetite for holding Argentinian sovereign debt issues (the asset of concern), despite the lack of any material direct connection between the Russian and Argentinian economies, because of the asset managers direct losses of capital due to the Russian default, and perhaps indirectly through demands for liquidation by relatively unsophisticated clients. Or, for example, when Hurricane Andrews struck, reinsurers with exposure to that event would have had a sudden reduction in capital available to cover monoline insurers facing, say, earthquake risk. While one could use the model of aversion to correlated endowment risk of Section 2 to compute the illiquidity loss rate δ associated with an aggregate shock, we would prefer to discuss the implications of the aggregate shock in more general terms, abstracting from the determination of the illiquidity loss rate δ. 8 The steady-state masses, absent new shocks, are µ lo = and µ ln =

24 8.5 8 Price Instantaneous Return Calendar Time Figure 4: The top panel shows the price as a function of time when an aggregate liquidity shock at time 0.4. The bottom panel shows the corresponding annualized realized returns. So, thinking of the aggregate shock as a simultaneous loss in capital to many investors that induces a reduced appetite by them for owning the asset in question, we may view an investor affected by the shock as having the intensity λ u for a recapitalizing event, after which that investor no longer has an elevated distress cost δ for owning the asset. The price and return dynamics associated with these parameters are shown in Figure 4. The top panel of the figure shows prices and the bottom panel shows realized instantaneous returns, both as functions of calendar time for a specific state of nature. At time t = 0.4, the economy experiences an aggregate liquidity shock, causing the asset price to drop suddenly by about 15%. Notably, it takes more than a year for the asset price to recover 23

25 to a roughly normal level. A buyer who was able to locate a seller immediately after the shock and realized an annualized return of roughly 30% for several months. While one is led to think in terms of the value to this vulture buyer of having retained excess liquidity so as to profit at times of aggregate liquidity shocks, our model has only one type of buyer, and is therefore too simple to address this sort of vulture specialization. The large illustrated price impact of a shock is due to the large number of sellers and the relatively low number of potential buyers that are available immediately after a shock. The roughly 50% reduction in potential buyers that occurred at the illustrated shock increased a seller s expected time to find a potential buyer from 6.2 days immediately before the shock to 12.4 days immediately after the shock. Further, once a seller finds a buyer, the seller s bargaining position is poor because of his reduced outside search options and the buyer s favorable outside options. Naturally, high-type owners who become low-type owners experience the largest utility loss from a shock. 9 The utility loss for low-type owners is also large, since their prospects of selling worsen significantly. High-type owners who do not themselves become low-type during the shock are not affected much since they expect the market to recover to normal conditions before they need to make a sale (given that λ d = 0.2 is small relative to the length of the recovery period). The agents who don t hold the asset when the shock hits are positively affected since they stand a good chance of being able to benefit from the sell-side pressure. The prospect of future aggregate liquidity shocks affects prices. For this, we compare the price a long time after a shock has hit (absent new shocks, that is, lim t P(t)) with the steady-state asset price, P ζ=0 = 9.25 associated with an economy with no aggregate shocks (ζ = 0), but otherwise the same parameters. The presence of aggregate liquidity shocks reduces the price, in this sense, by 12.5%. There are two channels through which the price responds after an aggregate shock. One is the search technology for trading. The other is the recovery of individual investors from the shock itself, captured by the switching intensities λ u and λ d. We have already motivated λ u as the mean rate at which a financially distressed investor can be recapitalized. 9 The utilities of the owners drop from V ho = 9.29 and V lo = 9.14, respectively, to V ho = 9.24 and V lo = 8.47, while the values of the non-owners increase from V hn = 1.13 and V ln = 1.10 to V hn = 2.22 and V ln =